In recent years, it has been observed that the average student enrolled in an introductory calculus course at the college level is not as adept as she once was in her prerequisite algebra/precalculus skills. Whether this comparison with the past is accurate or not, it is certainly true that the average student makes many errors of an algebraic (rather than a calculus) nature, and this serves only to divert her focus from the subject at hand.

To address the issue of poor algebraic skills, many writers of calculus texts now include a preliminary chapter for the review of precalculus concepts. While, in theory, such a review seems a good idea, in practice, the benefits, at least in the opinion of some, are dubious. In the ideal setting such a review would get ample time (with some students getting advised to take an entire precalculus course at the college level before entering calculus); instead, it is usually clear that the teacher is trying to spend the minimal time necessary so as to arrive at the calculus as early in the semester as possible.

This document was written, both out of experiences that have led to my abandoning an algebra review at the start of first semester calculus sections I teach, and out of certain realities at my institution. As to the former, it is my experience that a two-week review of precalculus is of little value. One might think that the stronger students might be bored for a couple of weeks while the struggling ones gratefully soak in the understanding of precalculus that has previously eluded them. In my experience, it works the other way around. The strong students -- the ones who would do relatively well with no review -- are the most engaged, while the weaker ones are buoyed up by a false confidence bred out of a sense of familiarity with the topics. For such students there are two rude awakenings to come at the end of the review: the score on the exam testing their knowledge of precalculus, and the very unfamiliar and difficult (to some more than others) concepts of calculus, now thrown more quickly at them because several weeks were given over to review. It is a discouraging way to begin a course, and many students never get their heads above water again during the semester. In addition to experiences such as these, I have a certain sympathy for the point of view that students who have enrolled in a calculus course should get just that, not some hybrid course that is inappropriate both for those who have strong algebra skills, and for those whose lack of algebraic facility calls for a more extensive study of precalculus.

As to the realities at my institution, recent changes have led to the addition, rather than elimination, of certain topics which had not formerly been in the first year of calculus. Roughly speaking, two and a half semesters have been compressed into two. Many of our students in first-year calculus are studying to be engineers, and these changes are the result of the attempts on the part of the Mathematics Department to accomodate the wishes of our Engineering Department, while maintaining a calculus sequence that is coherent and mathematically sound. As one might expect, we have tried to make the total course content for both semesters of calculus comparable to what it was before the change; we have increased breadth at the expense of depth. Nevertheless, several weeks of review at the start of a course seems a more remote possibility than before.

It is not my response to ignore the sometimes-incomplete backgrounds of our students, but to deal with the issue differently than with an initial precalculus review. I usually take a just-in-time approach, providing a short treatise on various precalculus concepts at the moments they become relevant in our discussion of calculus. For instance, I usually treat the subject of inverses about the time one discusses the chain rule, inverse trigonometric functions, or logarithms and exponentials. Still, the day-to-day algebraic errors which students make, such as the assumption that all functions behave linearly, are not directly addressed by a review of the most important precalculus concepts, whether this review comes all-at-once at the beginning of the course, or in short bursts interspersed among calculus topics. It is for these remaining pervasive errors that I have written this piece.

To be straight about it, its purpose is two-fold. One purpose is as mentioned above, to provide a detailed discussion of some of the most common algebraic errors. The other is to reduce the amount of writing a conscientious grader feels compelled to do. I achieve the latter by giving names and 3 or 4-letter acronyms to the error types. When one of these errors is seen, a grader need only write the acronym, not a long discussion of the error. The student who receives the abbreviated 4-letter comment is not shortchanged in the least. She may turn to this document, look up the acronym, read the accompanying discussion, and benefit from it just as much (more?) as she would have from a detailed comment.

Of course, for this grading/commenting system to be successful, students must have access to this document. It currently is available on the web at http://www.calvin.edu/~scofield/courses/materials/tae/.

Perhaps the bigger hurdle is that teachers must take the time to read and note which errors are addressed herein, along with the acronym used for each. It may take a nontrivial amount of time on the part of the instructor to get accustomed to all of these. Still, I believe the time invested in the beginning should, in the long run, reduce an instructor's grading time.

Please address comments on this document -- both those intended to improve discussion of errors already addressed and those indicating an error which you think should be addressed -- to the author. Also, if you use this document, a vote of confidence by way of a quick email would be appreciated.

Top Algebra Errors Made by Calculus Students (full document)

Full List of Grading Codes

Thomas L. Scofield 2003-09-04